Finite Group Databasebeta

← #6375747

Subgroups — #6375747

Order 512.

1996 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3077C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic, complemented
3#34058C211yesmaximal, characteristic
4#22126C214yesmaximal, complemented
5#22122C214yesmaximal, complemented
6#22117C212yesmaximal, complemented
7#22116C212yesmaximal, complemented
8#22111C212yesmaximal, complemented
9#22110C212yesmaximal, complemented
10#21311C211yesmaximal, characteristic, complemented
11#21171C211yesmaximal, characteristic, complemented
12#21160C212yesmaximal, complemented
13#21091C212yesmaximal
14#21063C212yesmaximal
15#21036C212yesmaximal, complemented
16C2×#1802C211yesmaximal, characteristic
17#20431C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#3077C4116yescomplemented
21C2×#478C2214yes—
22C2×#476C2212yes—
23C2×#474C2212yes—
24C2×#428C2212yes—
25C2×#428C2212yes—
26C2×#423C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C2×#418C2212yes—
29C22×#102C2212yes—
30C2×#405C2212yes—
31#2360—22no—
32#2360—22no—
33#2355—21no—
34#2355—21no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4—21no—
37#2330C2214yescomplemented
38#2319—21no—
39#2319—21no—
40#2319C2212yes—
41#2319C2212yes—
42#2317C2212yes—
43#2316C2211yescharacteristic
44#2316C2211yescharacteristic
45#2312—22no—
46#2309C2211yescharacteristic
47#2309C2211yescharacteristic
48C4×C4⋊Q8C2212yescomplemented
49C4×C4⋊Q8C2212yescomplemented
50#2293—21no—
51#2293—21no—
52#2293C2212yes—
53#2293C2212yes—
54C2×#384C2211yescharacteristic
55#2280—22no—
56#2274C2212yes—
57C4×#117C2211yescharacteristic
58C4×#117C2211yescharacteristic
59#1989C2212yescomplemented
60#1989C2212yescomplemented
61#1988C2211yescharacteristic, complemented
62#1988C2211yescharacteristic, complemented
63#1987C2211yescharacteristic, complemented
64#1987C2211yescharacteristic, complemented
65#1948C2212yescomplemented
66#1948C2212yescomplemented
67#1947C2212yescomplemented
68#1947C2212yescomplemented
69#1944C2214yescomplemented
70#1943C2214yescomplemented
71#1942C2214yescomplemented
72#1941C2214yescomplemented
73#1936C2214yes—
74#1936C2214yes—
75#1934C2214yes—
76#1934C2214yes—
77#1926C2214yes—
78#1926C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1921C2214yescomplemented
82#1920C2214yescomplemented
83#1919C2214yescomplemented
84#1918C2214yescomplemented
85#1802C2212yes—
86#1802C2212yes—
87#1802C2214yes—
88#1768C2214yescomplemented
89#1767C2212yescomplemented
90C4×C4⋊C8C2212yescomplemented
91#1736C2212yes—
92#1735C2212yes—
93#1735C2212yes—
94C2×#340C2212yes—
95#1729C2212yes—
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C22×M16C2312yes—
99C23×C8C2312yes—
100#533—24no—
101#533C2×C418yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125C2312yes—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4C2×C418yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×#117—22no—
118C2×#117—22no—
119C2×#117C2311yescharacteristic
120C2×#117C2311yescharacteristic
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2311yescharacteristic
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C42C2311yescharacteristic
130#478C2×C4132yescomplemented
131#476C2×C4116yescomplemented
132#474C2×C4116yescomplemented
133#428—44no—
134#428C2×C4116yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106C2314yes—
138C2×#106C2314yes—
139#423C2×C418yes—
140C2×C4⋊C8—24no—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146#418C2×C4116yes—
147C2×#102—44no—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#405—44no—
153C2×#97C2314yes—
154C2×C4×C8C2314yes—
155#391—22no—
156#391—22no—
157#390—21no—
158#390—21no—
159#390—21no—
160#390—21no—
161#390C2312yes—
162#390C2312yes—
163#390C2312yes—
164#390C2312yes—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389—24no—
172#389—24no—
173#389C2314yes—
174#389C2314yes—
175#389C2312yes—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#388—22no—
187#387—24no—
188#387—24no—
189#385—24no—
190#384—22no—
191#384—21no—
192#384—21no—
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#381—22no—
197#381—22no—
198#380—22no—
199#380—22no—
200#380—22no—
201#380—22no—
202#379—22no—
203#379—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C22⋊C4—21no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—24no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4D412yes—
225#376—21no—
226#376C2312yes—
227C43—21no—
228C43C2312yes—
229#340C2314yes—
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2312yes—
239#340C2312yes—
240#340C2312yes—
241#336C2314yes—
242#336C2314yes—
243C25—21no—
244C22×Q8—21no—
245C22×Q8—21no—
246C22×D4—21no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—22no—
252C22×D4C22×C414yes—
253C22×D4C2412yes—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—28no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4—82no—
260C23×C4—82no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2411yescharacteristic
264C2×M16—24no—
265C2×M16—24no—
266C2×M16C22×C418yes—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C4⋊Q8—22no—
274C4⋊Q8—22no—
275C4⋊Q8C22×C414yes—
276C4⋊Q8C22×C414yes—
277#125—24no—
278#125C22×C418yes—
279C22⋊Q8—28no—
280C22⋊Q8—28no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—84no—
284C4×Q8—84no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—24no—
288C4×D4—24no—
289C4×D4C22×C418yes—
290C4×D4C22×C418yes—
291#117—24no—
292#117—22no—
293#117—22no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—22no—
299#117—22no—
300#117—84no—
301#117—84no—
302#117C22×C418yes—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4—82no—
324C2×C4⋊C4C22×C414yes—
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4C2×D411yescharacteristic
328C2×C4⋊C4C2×D411yescharacteristic
329C2×C4⋊C4C2414yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2411yescharacteristic
333C2×C4⋊C4C2411yescharacteristic
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—28no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—22no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42C22×C414yes—
363C2×C42C2×D412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42#4114yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—44no—
375#107C22×C4116yes—
376#106—44no—
377#106C22×C4116yes—
378C4⋊C8—24no—
379C4⋊C8—22no—
380C4⋊C8—22no—
381C4⋊C8C22×C418yes—
382C4⋊C8C22×C418yes—
383#102—44no—
384#102—44no—
385#102C22×C4116yes—
386#102C22×C4116yes—
387#97—44no—
388C4×C8—44no—
389#95—21no—
390#95—21no—
391#95—21no—
392#95—21no—
393#95—24no—
394#95—24no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—24no—
400#95—24no—
401#95C2×D412yes—
402#95C2×D412yes—
403#95C2×D412yes—
404#95C2×D412yes—
405C24—22no—
406C24—24no—
407C24—42no—
408C24—42no—
409C24—81no—
410C24—81no—
411C24—81no—
412C24—81no—
413C24C22×D411yescharacteristic
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—42no—
417C2×Q8—42no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×D4—24no—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4C23×C414yes—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—28no—
440C22×C4—28no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—816no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4C4×D414yes—
459C22×C4C4×D414yes—
460C22×C4C23×C412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)12yes—
472C22×C42−1+412yes—
473C22×C42−1+412yes—
474C22×C42−1+412yes—
475C22×C4C2511yescharacteristic
476M16—216no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—48no—
486C2×C8C23×C418yes—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—24no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4—82no—
518C4⋊C4—84no—
519C4⋊C4—84no—
520C4⋊C4C23×C414yes—
521C4⋊C4C23×C414yes—
522C4⋊C4C23×C414yes—
523C22⋊C4—28no—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—28no—
527C22⋊C4—28no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C22⋊C4—82no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C42—28no—
539C42—24no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—42no—
551C42—81no—
552C42—81no—
553C42—84no—
554C42—84no—
555C42C4×D414yes—
556C42C4×D414yes—
557C42C23×C412yes—
558C42C23×C412yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—24no—
566C23—42no—
567C23—42no—
568C23—48no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—84no—
574C23—84no—
575C23—84no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23C2×C4×D412yes—
580C23#53612yes—
581C23#53712yes—
582C23C23×D411yescharacteristic
583Q8—48no—
584Q8—48no—
585D4—24no—
586D4—24no—
587D4—48no—
588D4—42no—
589D4—42no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—416no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—88no—
636C2×C4—84no—
637C2×C4—164no—
638C2×C4C2×C4×D412yes—
639C2×C4C2×C4×D412yes—
640C2×C4C2×C4×D412yes—
641C2×C4C2×C4×D412yes—
642C2×C4C2×C4×D412yes—
643C2×C4#51914yes—
644C2×C4#52014yes—
645C2×C4#53612yes—
646C2×C4#53712yes—
647C2×C4#54114yes—
648C2×C4#54214yes—
649C2×C4#57612yes—
650C2×C4#57712yes—
651C2×C4C24×C411yescharacteristic
652C2×C4C22×#4111yescharacteristic
653C2×C4C2×H(5,𝔽2)11yescharacteristic
654C2×C4C2×2−1+411yescharacteristic
655C2×C4C2×2−1+411yescharacteristic
656C2×C4C2×2−1+411yescharacteristic
657C8—28no—
658C8—42no—
659C8—42no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—816no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22#232314yes—
675C22C22×C4×D411yescharacteristic
676C22C2×#53611yescharacteristic
677C22C2×#53711yescharacteristic
678C22C2×#57611yescharacteristic
679C22C2×#57711yescharacteristic
680C22#358511yescharacteristic
681C22#358611yescharacteristic
682C4—22no—
683C4—22no—
684C4—22no—
685C4—44no—
686C4—42no—
687C4—42no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—84no—
696C4—81no—
697C4#232314yes—
698C4#294614yes—
699C4C2×#51911yescharacteristic
700C4C2×#52011yescharacteristic
701C4C2×#54111yescharacteristic
702C4C2×#54211yescharacteristic
703C2—24no—
704C2—84no—
705C2C2×#232311yescharacteristic
706C2C2×#294611yescharacteristic
707C2#6017011yescharacteristic
708C2#6088811yescharacteristic
709C2#6089211yescharacteristic
710C2#6092211yescharacteristic
711C2#6092311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #34214C2semidirect
4. #22126C2semidirect
5. #22122C2semidirect
6. #22117C2semidirect
7. #22116C2semidirect
8. #22111C2semidirect
9. #22110C2semidirect
10. #21311C2semidirect
11. #21171C2semidirect
12. #21160C2semidirect
15. #21036C2semidirect
17. #20431C2semidirect
20. #3077C4semidirect
37. #2330C22semidirect
48. C4×C4⋊Q8C22semidirect
49. C4×C4⋊Q8C22semidirect
59. #1989C22semidirect
60. #1989C22semidirect
61. #1988C22semidirect
62. #1988C22semidirect
63. #1987C22semidirect
64. #1987C22semidirect
65. #1948C22semidirect
66. #1948C22semidirect
67. #1947C22semidirect
68. #1947C22semidirect
69. #1944C22semidirect
70. #1943C22semidirect
71. #1942C22semidirect
72. #1941C22semidirect
81. #1921C22semidirect
82. #1920C22semidirect
83. #1919C22semidirect
84. #1918C22semidirect
88. #1768C22semidirect
89. #1767C22semidirect
90. C4×C4⋊C8C22semidirect
130. #478C2×C4semidirect
131. #476C2×C4semidirect
132. #474C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#30772
#342142
#340582
#221262
#221222
#221172
#221162
#221112
#221102
#213112
#211712
#211602
#210912
#210632
#210362
C2×#18022
#204312